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\usepackage{amsmath, amssymb} % 数学公式与符号
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\title{实变函数第一章：集合}
\author{CQX ET AL}

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\begin{document}

\begin{frame}
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\begin{frame}{第一章目录 }

\begin{enumerate}

\item[1.1.] 集合的表示
\item[1.2.] 集合的运算：德摩根公式。集合序列的上下极限。
\item[1.3.] 对等和基数：伯恩斯坦定理。并集、直积与幂集的基数。
\item[1.4.] 可数集合：可数基数与连续基数。
\item[1.5.] 不可数集合

\end{enumerate}

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%\begin{frame}{第一章重点 }
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\begin{frame}{1.1.1. 集合的表示 }

\begin{itemize}

\item  {\color{red}问题：什么是集合？什么是元素？ }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.1.2.  }

\begin{itemize}

\item  {\color{red}问题：用集合的语言描述函数的性质： }
\begin{enumerate}
\item  {\color{red}函数 $f(x)$ 在实数轴上有定义，且在 $[a,b]$ 上有上界 $M$.  }
\item  {\color{red}函数 $f(x)$ 在实数轴上有定义，且在点 $x_0\in\mathbb{R}$ 连续。 }
\item  {\color{red}函数 $f(x)$ 在闭区间 $[a,b]$ 上有定义，其最大值和最小值分别为 $m$ 和 $M$.  }
\end{enumerate}


%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.1.3.  }

\begin{itemize}

\item  {\color{red}问题：符号 $\mathbb{N}, \mathbb{Z},  \mathbb{Q},  \mathbb{R},  \mathbb{C}$ 的含义分别是什么？}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.2.1. 集合的运算 }

\begin{itemize}

\item  {\color{red}问题：设有一族集合 $\{A_\alpha: \alpha\in\Lambda\}$, 其中 $\Lambda$ 是一个指标集。
写出这族集合的并集的定义。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.2.2.  }

\begin{itemize}

\item  {\color{red}问题：证明下述结论：}\vspace{0.3cm}

\begin{enumerate}
\item  {\color{red}设 $E\subseteq\mathbb{R}$, 设 $f(x)$ 和 $g(x)$ 是定义在 $E$ 上的函数，则对任意 $c\in\mathbb{R}$, 成立 $$\{x\in E: \max (f(x),g(x)) > c\} = \{x\in E: f(x)>c\} \cup \{x\in E: g(x)>c\}. $$ } 

\item  {\color{red}对任意两个实数 $a<b$, 成立 $$(a,b) = \bigcup\limits_{n=1}^\infty \left[ a+\frac{1}{n}, b-\frac{1}{n} \right]. $$ }

\end{enumerate}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.2.3.  }

\begin{itemize}

\item  {\color{red}问题：证明下述结论：}
\begin{enumerate}

\item  {\color{red} 记 $\mathbb{Q}_n = \{\frac{m}{n}: m\in\mathbb{Z}\}, n=1,2,\cdots$, 
则有 $$\mathbb{Q} = \bigcup\limits_{n=1}^{\infty} \mathbb{Q}_n.$$ } 

\item  {\color{red} 设 $E\subseteq\mathbb{R}$, 设 $f(x)$ 是定义在 $E$ 上的函数，则有
$$E(f>0) := \{x\in E: f(x)>0\} = \bigcup\limits_{n=1}^{\infty} \left\{ x\in E: f(x)>\frac{1}{n} \right\}. $$ }


\end{enumerate}

%\item  解答：


\end{itemize}

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\begin{frame}{1.2.4.  }

\begin{itemize}

\item  {\color{red}问题：设有一族集合 $\{A_\alpha: \alpha\in\Lambda\}$, 其中 $\Lambda$ 是一个指标集。
写出这族集合的交集的定义。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.2.5.   }

\begin{itemize}

\item  {\color{red}问题：证明下述结论：}
\begin{enumerate}
\item  {\color{red}设 $f(x)$ 是定义在 $E$ 上的函数，则有 $$\{x\in E: a<f(x)\le b\} = \{x\in E: f(x)>a\}\cap \{x\in E: f(x)\le b\}. $$ } 

\vspace{-0.3cm}

\item  {\color{red}设 $\{f_n(x):n=1,2,\cdots\}$ 是定义在 $E$ 上的一列函数，则对任意实数 $c$, 有
\begin{eqnarray*}
\{x\in E: \sup \{f_n(x)\} \le c\} &=& \bigcap\limits_{n=1}^\infty \{x\in E: f_n(x)\le c\}, \\ 
\{x\in E: \sup \{f_n(x)\} > c\} &=& \bigcup\limits_{n=1}^\infty \{x\in E: f_n(x)> c\}. 
\end{eqnarray*}
}

\end{enumerate}

\vspace{-0.3cm}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.2.6.  }

\begin{itemize}

\item  {\color{red}问题：}
\begin{enumerate}
\item  {\color{red}设 $A,B$ 是两个集合，什么是 $A$ 和 $B$ 的差集？ } 
\item  {\color{red}设集合 $A$ 是全集 $S$ 的一个子集，什么是 $A$ 的补集？} 
\end{enumerate}


%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.2.7.  }

\begin{itemize}

\item  {\color{red}问题：证明德摩根公式：设 $\{A_\alpha:\alpha\in\Lambda\}$ 是全集 $S$ 的一族子集，则有
\begin{eqnarray*}
S - \bigcup\limits_{\alpha\in \Lambda} A_\alpha &=& \bigcap\limits_{\alpha\in \Lambda} (S - A_\alpha), \\ 
S - \bigcap\limits_{\alpha\in \Lambda} A_\alpha &=& \bigcup\limits_{\alpha\in \Lambda} (S - A_\alpha).  
\end{eqnarray*}

 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.2.8.  }

\begin{itemize}

\item  {\color{red}问题：设 $\{f_n(x)\}$ 是定义在 $E$ 上的函数列。固定 $x\in E$. 若存在正数 $M>0$ 使得对任意 $n$ 都有 $|f_n(x)|\le M$, 则称数列 $\{f_n(x)\}$ 是有界的。将这个条件写成一些集合的交集与并集的形式。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.2.9.  }

\begin{itemize}

\item  {\color{red}问题：设 $\{f_n(x):n\in\mathbb{N}\}$ 是定义在 $E$ 上的函数列。则有
$$\{x\in E: \lim\limits_{n\to\infty} f_n(x) = 0\} = \bigcap\limits_{\varepsilon\in\mathbb{R}^+} 
\bigcup\limits_{N=1}^{\infty} \bigcap\limits_{n=N}^{\infty} \{x\in E: |f_n(x)|<\varepsilon\}. $$
 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.2.10.  }

\begin{itemize}

\item  {\color{red}问题：}
\begin{enumerate}
\item  {\color{red}什么是集合序列 $A_1,A_2,\cdots,A_n,\cdots$ 的上极限和下极限？ }
\item  {\color{red}证明可以使用交集和并集来表示集合序列的上下极限：
\begin{eqnarray*}
\varlimsup\limits_{n\to\infty} A_n &=&  \bigcap\limits_{n=1}^{\infty} \bigcup\limits_{m=n}^{\infty} A_m, \\ 
\varliminf\limits_{n\to\infty} A_n &=&  \bigcup\limits_{n=1}^{\infty} \bigcap\limits_{m=n}^{\infty} A_m. 
\end{eqnarray*}
}
\end{enumerate}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.2.11.  }

\begin{itemize}

\item  {\color{red}问题：设 $A_n$ 是下述点集，
\begin{eqnarray*}
A_{2m+1} &=& \left[ 0,2-\frac{1}{2m+1} \right], m=0,1,2,\cdots, \\ 
A_{2m} &=& \left[ 0,1+\frac{1}{2m} \right], m=1,2,3,\cdots.
\end{eqnarray*}
计算集合序列 $A_1,A_2,\cdots,A_n,\cdots$ 的上极限和下极限。

 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.2.12.  }

\begin{itemize}

\item  {\color{red}问题：设 $f(x)$ 是定义在 $E$ 上的有限函数，记 $F_n = \{x\in E: |f(x)|\ge \frac{1}{n}\}$.
证明 $\{F_n\}$ 是增加集列，且有 
$$\lim\limits_{n\to\infty} F_n = \bigcup\limits_{n=1}^{\infty} F_n = \{x\in E: f(x)\neq 0\}. $$

 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.2.13.   }

\begin{itemize}

\item  {\color{red}问题： 设 $f(x)$ 是定义在 $E$ 上的有限函数，记 $E_n = \{x\in E: |f(x)|>n\}$.
证明 $\{E_n\}$ 是减少集列，且有 
$$\lim\limits_{n\to\infty} E_n = \bigcap\limits_{n=1}^{\infty} E_n = \{x\in E: f(x)=\infty\} = \varnothing. $$

 }


%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.3.1. 对等和基数 }

\begin{itemize}

\item  {\color{red}问题：什么时候称两个集合是对等的? } 
%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.3.2.  }

\begin{itemize}

\item  {\color{red}问题：使用对等的概念定义有限集。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.3.3.  }

\begin{itemize}

\item  {\color{red}问题：构造下述集合之间的对等： }
\begin{enumerate}
\item  {\color{red}正奇数全体与正偶数全体。 }
\item  {\color{red}正整数全体与正偶数全体。 }
\item  {\color{red}区间 $(0,1)$ 与全体实数。 }

\end{enumerate}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.3.4.  }

\begin{itemize}

\item  {\color{red}问题：证明对等关系是一种等价关系，即满足自反性、对称性和传递性。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.3.5.  }

\begin{itemize}

\item  {\color{red}问题：什么时候称两个集合有相同的基数？什么时候称一个集合的基数小于另一个集合的基数？ }

%\item  解答：


\end{itemize}

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\begin{frame}{1.3.6.   }

\begin{itemize}

\item  {\color{red}问题：证明伯恩斯坦定理：设 $A,B$ 是两个非空集合。如果 $A$ 对等于 $B$ 的一个子集，$B$ 也对等于 $A$ 的一个子集，那么 $A$ 与 $B$ 对等。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.4.1. 可数集合  }

\begin{itemize}

\item  {\color{red}问题：什么是可数集合或可列集合？ }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.4.2.   }

\begin{itemize}

\item  {\color{red}问题：证明：任意无限集合都包含一个可数子集。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.4.3.   }

\begin{itemize}

\item  {\color{red}问题：证明：可数集合的任何子集或者是有限集，或者是可数集。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.4.4.   }

\begin{itemize}

\item  {\color{red}问题：证明：设 $A$ 是可数集，$B$ 是有限集或者可数集，则 $A\cap B$ 为可数集。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.4.5.   }

\begin{itemize}

\item  {\color{red}问题：证明：可数个可数集的并集仍是可数集。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.4.6.   }

\begin{itemize}

\item  {\color{red}问题：证明：有理数全体是一个可数集。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.4.7.   }

\begin{itemize}

\item  {\color{red}问题：证明：设 $A,B$ 都是可数集，则 $A\times B$ 也是可数集。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.4.8.   }

\begin{itemize}

\item  {\color{red}问题：证明：整系数多项式全体是一个可数集。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.5.1. 不可数集合  }

\begin{itemize}

\item  {\color{red}问题：证明：实数全体是一个不可数集合。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.5.2.   }

\begin{itemize}

\item  {\color{red}问题：设 $a<b$ 是两个实数。证明区间 $(a,b)$ 是一个不可数集合。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.5.3.   }

\begin{itemize}

\item  {\color{red}问题：什么是连续基数？符号 $\aleph_0$ 与 $\aleph$ 的含义是什么？ }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.5.4.   }

\begin{itemize}

\item  {\color{red}问题：证明：设 $A_1,A_2,\cdots,A_n,\cdots$ 是一列互不相交的集合，设它们的基数都是 $\aleph$. 证明 $\cup_{n=1}^{\infty} A_n$ 的基数也是 $\aleph$.  }

%\item  解答：


\end{itemize}

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\begin{frame}{1.5.5.   }

\begin{itemize}

\item  {\color{red}问题：证明：设 $A_1,A_2,\cdots,A_n,\cdots$ 是一列基数都是 $\aleph$ 的集合。
证明乘积集合 $\prod_{n=1}^{\infty} A_n$ 的基数也是 $\aleph$. }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.5.6.   }

\begin{itemize}

\item  {\color{red}问题：证明：$\mathbb{R}^n$ 的基数也是 $\aleph$. }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.5.7.   }

\begin{itemize}

\item  {\color{red}问题：设 $M$ 是一个集合，记 $P(M)$ 是 $M$ 的幂集合，即 $M$ 的所有子集组成的集合。证明：$P(M)$ 的基数大于 $M$ 的基数。 }

%\item  解答：


\end{itemize}

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\begin{frame}{1.5.8.   }

\begin{itemize}

\item  {\color{red}问题：设 $f(x)$ 是定义在点集 $E$ 上的函数，则
$$\{x: f(x)=0\} = \bigcap\limits_{n=1}^{\infty}\left\{ x:  |f(x)|<\frac{1}{n}\right\} $$
}

%\item  解答：


\end{itemize}

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\begin{frame}{1.6.1. 习题1 }

\begin{itemize}

\item  {\color{red}问题：证明：}
\begin{enumerate}%[label={(\arabic*)}]
\item  {\color{red} $(A\backslash B)\backslash C = A\backslash (B\cup C)$. }
\item  {\color{red} $(A\cup B)\backslash C = (A\backslash C) \cup (A\backslash B)$. }
\end{enumerate}

%\item  解答：


\end{itemize}

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\begin{frame}{1.6.2. 习题4  }

\begin{itemize}

\item  {\color{red}问题：设 $A_{2n-1}=(0,1/n), A_{2n}=(0,n), n=1,2,\cdots$. 求集合序列 $\{A_n\}$ 的上限集合与下限集合。 }

%\item  解答：


\end{itemize}

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\begin{frame}{1.6.3. 习题7  }

\begin{itemize}

\item  {\color{red}问题：设 $f(x),g(x)$ 是定义在集合 $E$ 上的函数。证明：
$$\{x: f(x)>g(x) \} = \bigcup\limits_{n=1}^{\infty} \left\{ x: f(x) > g(x) + \frac{1}{n} \right\}. $$ 
}

%\item  解答：


\end{itemize}

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\begin{frame}{1.6.4. 习题10  }

\begin{itemize}

\item  {\color{red}问题：设 $\{f_n(x)\}$ 是定义在 $E$ 上的一列函数，设 $c$ 是任意实数。证明：
$$\{x: \underset{n}\inf \{f_n(x)\} \ge c \} = \bigcap\limits_{n=1}^{\infty} \left\{ x: f_n(x) \ge c \right\}. $$ 
}
%\item  解答：


\end{itemize}

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\begin{frame}{1.6.5. 习题14  }

\begin{itemize}

\item  {\color{red}问题：证明：球面上的点集与平面上的点集是对等的。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{1.6.6. 习题17  }

\begin{itemize}

\item  {\color{red}问题：证明：增函数的不连续点最多只有可数个。 }

%\item  解答：


\end{itemize}

\end{frame}



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